Simplifying Radicals An In-Depth Guide To 3√(8n³) ⋅ 4√(2n²)
In the realm of mathematics, simplifying radical expressions is a fundamental skill. In this article, we'll delve into the process of simplifying the expression 3√(8n³) ⋅ 4√(2n²), breaking down each step to ensure clarity and understanding. This comprehensive guide will not only provide the solution but also equip you with the knowledge to tackle similar problems with confidence.
Understanding Radicals
Before we dive into the simplification process, it's essential to grasp the concept of radicals. A radical expression consists of a radical symbol (√), a radicand (the expression under the radical), and an index (the small number indicating the root to be taken). For instance, in the expression √(9), the radical symbol is √, the radicand is 9, and the index is 2 (since it's a square root). Understanding these components is crucial for manipulating and simplifying radical expressions effectively.
Breaking Down the Expression
Our given expression is 3√(8n³) ⋅ 4√(2n²). To simplify it, we'll employ several key strategies, including factoring, applying the product rule for radicals, and extracting perfect squares. Let's begin by examining the individual terms and identifying any factors that are perfect squares. This initial step sets the stage for the subsequent simplification process.
Factoring and Identifying Perfect Squares
The first step in simplifying 3√(8n³) ⋅ 4√(2n²) involves factoring the radicands (the expressions under the radical signs) to identify perfect square factors. This process allows us to extract these perfect squares from the radicals, thereby simplifying the overall expression. Let's break down the radicands:
- 8n³: We can factor 8 as 2 * 2 * 2, which can be written as 2² * 2. Additionally, n³ can be expressed as n² * n. Thus, 8n³ can be rewritten as 2² * 2 * n² * n.
- 2n²: Here, 2 is a prime number and cannot be factored further. However, n² is already a perfect square.
By identifying these perfect square factors (2² and n²), we pave the way for simplifying the radicals in the next steps. This approach makes it easier to extract the square roots and reduce the complexity of the expression.
Applying the Product Rule for Radicals
The product rule for radicals is a fundamental principle that simplifies the multiplication of radicals with the same index. This rule states that the square root of a product is equal to the product of the square roots, mathematically expressed as √(ab) = √a ⋅ √b. Applying this rule to our expression, 3√(8n³) ⋅ 4√(2n²), allows us to combine the terms under a single radical, making the simplification process more streamlined.
Combining Terms Under a Single Radical
Before applying the product rule, it's helpful to rewrite the expression by grouping the constants and the radicals separately:
(3 ⋅ 4) ⋅ (√(8n³) ⋅ √(2n²))
This simplifies to:
12 ⋅ (√(8n³) ⋅ √(2n²))
Now, we can apply the product rule by multiplying the radicands together:
12 ⋅ √(8n³ ⋅ 2n²)
This step combines the two separate radicals into a single radical, setting the stage for further simplification by multiplying the terms under the radical.
Multiplying the Radicands
Inside the radical, we now have the expression 8n³ ⋅ 2n². To simplify this, we multiply the coefficients (8 and 2) and add the exponents of the variable n:
8n³ ⋅ 2n² = (8 ⋅ 2) ⋅ (n³ ⋅ n²)
This yields:
16n⁵
So, our expression now looks like this:
12 ⋅ √(16n⁵)
This consolidation of terms under a single radical and the multiplication of the radicands simplifies the expression significantly, bringing us closer to the final simplified form.
Extracting Perfect Squares
To further simplify the expression 12 ⋅ √(16n⁵), we need to extract any perfect square factors from the radicand (16n⁵). This involves identifying factors that have integer square roots and pulling them out of the radical. By doing so, we can reduce the expression to its simplest form.
Identifying Perfect Square Factors in 16n⁵
Let's break down 16n⁵ into its factors:
- 16: This is a perfect square, as 16 = 4².
- n⁵: We can rewrite n⁵ as n⁴ ⋅ n. Here, n⁴ is a perfect square because it can be expressed as (n²)².
Thus, we can rewrite 16n⁵ as 4² ⋅ (n²)² ⋅ n.
Extracting the Square Roots
Now that we've identified the perfect square factors, we can extract their square roots from the radical:
√(16n⁵) = √(4² ⋅ (n²)² ⋅ n)
Using the property √(a²)=a, we get:
4n²√n
Substituting this back into our expression, we have:
12 ⋅ (4n²√n)
This extraction of perfect squares from the radical is a crucial step in simplifying radical expressions, allowing us to reduce the expression to its most manageable form.
Final Simplification
After extracting perfect squares, we're now at the final stage of simplifying the expression 12 ⋅ (4n²√n). This involves multiplying the terms outside the radical to obtain the ultimate simplified form. Let's carry out the multiplication:
Multiplying the Terms
We have 12 multiplied by 4n²√n. To simplify this, we multiply the coefficients (12 and 4):
12 ⋅ 4n²√n = (12 ⋅ 4) ⋅ n²√n
This yields:
48n²√n
Final Simplified Expression
Therefore, the simplified form of the original expression 3√(8n³) ⋅ 4√(2n²) is:
48n²√n
This final result represents the expression in its most reduced and understandable form, where all possible simplifications have been made.
Matching with the Given Options
Now that we have the simplified expression, we can compare it with the options provided:
A) 4√30 B) 16 C) 48n²√n D) 4 E) √10
Our simplified expression, 48n²√n, matches option C.
Conclusion
In this article, we've walked through the step-by-step process of simplifying the radical expression 3√(8n³) ⋅ 4√(2n²). We began by understanding the fundamentals of radicals, then broke down the expression, applied the product rule for radicals, and extracted perfect squares. The final simplified form is 48n²√n, which corresponds to option C. Mastering these techniques will empower you to simplify a wide range of radical expressions with ease and accuracy. This detailed guide provides not just the answer, but a thorough understanding of the methods involved, ensuring you're well-equipped to tackle similar problems in the future.
